Miguel Piñar

Prof Miguel Piñar is a Professor at the University of Granada. His research focuses on approximation theory and its connections with special functions, particularly orthogonal polynomials. He has made important contributions to the study of orthogonal Sobolev polynomials and their algebraic, differential, and asymptotic properties. Since 2003, his work has expanded to multivariate orthogonal polynomials, where he investigates their complexity using a matrix-based approach and extends classical results to domains beyond standard geometries.

Multivariate orthogonal polynomials and the numerical solution of PDE

Orthogonal polynomials play a central role in the numerical approximation of partial differential equations (PDEs), forming the foundation of spectral and high-order methods. While classical orthogonal polynomials are typically defined with respect to standard inner products, many problems arising in the variational formulation of PDEs naturally involve derivatives, motivating the use of Sobolev inner products. This leads to the theory of Sobolev orthogonal polynomials, which incorporates derivative information directly into their structure. In this talk, we explore recent developments in the theory of multivariate orthogonal polynomials with an emphasis on Sobolev orthogonality. We discuss how these polynomial systems can be constructed and analyzed in multiple dimensions, highlighting both their algebraic properties and computational challenges. Particular attention is given to their role in the design of efficient and accurate numerical schemes for PDEs, including their use in spectral methods. We also examine how Sobolev orthogonal polynomials can improve stability and convergence in problems where boundary conditions and derivative constraints play a significant role. Examples and applications will illustrate how these ideas contribute to advancing high-order numerical methods for PDEs.

Stefano De Marchi

Prof Stefano De Marchi is a Professor at the University of Padua, Italy, recognised for his contributions to approximation theory, multivariate polynomial and radial basis function approximation, and rational interpolation. He was an Erskine Fellow at the University of Canterbury in 2018. More recently, in 2025, he was awarded a grant from Sweden’s Knut and Alice Wallenberg Foundation to serve as a Visiting Professor at Uppsala University’s Department of Information Technology for conducting research in computational simulation and rational approximation methods.

From Radial Basis Functions to VS(D,R) kernels

Radial Basis Functions (RBF) have become quite a popular tool in mathematics and applied sciences, for their flexibility, allowing us to work with multidimensional data, without requiring any special structure of the data to be approximated. Usually, RBF depends on a shape parameter that needs to be tuned or optimised. Variably Scaled Kernels, or their counterparts for discontinuous functions, called Variably Scaled Discontinuous Kernels or the Variably Scaled Rational Kernels, are known alternatives that use a scaling function that allows for better mimicking the data. In this talk, after a review of the most important properties of RBF approximations, we present VS(D, R) kernels and some of their applications.

Gerlind Plonka-Hoch

Prof Gerlind Plonka-Hoch is a Professor of Applied Mathematics at the University of Goettingen, Germany. Her research spans numerical Fourier analysis, approximation and wavelet theory, regularisation methods, fast algorithms, and their applications in signal and image processing. She has been recognised with the Heinz-Maier-Leibnitz Prize and was selected as the Emmy Noether Lecturer by the German Mathematical Society in 2016.

EXPOWER: Methods for reconstruction of parametric exponential models

We consider the parametric exponential model

\[ f(t) = \sum_{j=1}^{M} \gamma_{j} \, \mathrm{e}^{\phi_{j}t} = \sum_{j=1}^{M} \gamma_{j} \, z_{j}^{t}, \]

where \(M \in \mathbb{N}\), \(\gamma_{j} \in \mathbb{C}\setminus\{0\}\), and \(z_{j} = \mathrm{e}^{\phi_{j}} \in \mathbb{C}\setminus\{0\}\) with \(\phi_{j} \in \mathbb{C}\) are pairwise distinct. The recovery of such exponential sums from a finite set of possibly corrupted signal samples plays an important role in many signal processing applications, as in phase retrieval, signal approximation, sparse deconvolution in nondestructive testing, model reduction in system theory, direction of arrival estimation, exponential data fitting, or reconstruction of signals with finite rate of innovation. Models of the above form are closely related to other parametric function models, as e.g. \(f(t) = \sum_{j=1}^{M} \gamma_{j} \phi(t-T_{j})\) with a given basis function \(\phi\) and arbitrary \(T_{j} \in \mathbb{R}\), or \(f(t) = \sum_{j=1}^{M} \gamma_{j} \cos(2\pi a_{j} t + b_{j})\).

There is however also a close relation to rational approximation models of the form \(f(z) = \sum_{j=1}^{N} \frac{a_{j}}{t - b_{j}}\).

All parameters of these models can be recovered using the Prony method or its generalizations. However, to improve the stability of the reconstruction, we want to exploit the strong connections between the algebraic methods and other approaches from rational approximation, structured matrix theory and optimization. This talk gives a survey on different numerical approaches to the recovery problem for exponential models.

Bernhard Beckermann

Bernhard Beckermann is Professor of Applied Mathematics at the University of Lille. His research bridges approximation theory and computational methods, with a focus on constructive approximation in the complex plane (particularly rational and orthogonal approximation), numerical linear algebra (including structured matrices, Krylov methods, and matrix functions), and numerical analysis, particularly algorithmic error estimation and the conditioning of polynomial and rational bases.

Rational approximation of Markov functions

The aim of this talk is to give new explicit upper bounds for the relative error of best rational approximents of Markov functions (that is, the Cauchy transform of some measure \(\mu\) with real support). In the special case of the Markov function \(f(z)=1/\sqrt{z}\), explicit formulas and bounds have been given already by Zolotarev more then 100 years ago, but are still subject of interest, see for instance a recent paper of Braess and Hackbusch. We will show that, within the class of measures satisfying the Szegő condition, it is known that, asymptotically, the worst case measure with support in \([a,b]\) is the equilibrium measure. For general measures on \([a,b]\) such like discrete measures, we show that there is a worst case measure, and thus we get new sharp upper bounds depending only on the capacity of the underlying condenser, and the degree of the rational function.

Stefan Güttel

Stefan Güttel is Professor of Applied Mathematics at the University of Manchester. His research focuses on computational mathematics, including numerical algorithms for large-scale linear algebra problems. A significant part of his work involves rational approximation, particularly through rational Krylov methods and algorithms such as RKFIT for computing rational approximants, with applications in scientific computing. His contributions have been recognized with the 2021 SIAM James H. Wilkinson Prize in Numerical Analysis and Scientific Computing and the 2023 ILAS Taussky-Todd Prize.

Shared-denominator rational approximation

Several problems in scientific computing can be related to approximating a family of parameter-dependent scalar functions, say \(f^\tau(z)\), by rational functions \(r^\tau(z)\) that all share the same denominator. Of particular interest are the resolvent function \(f^\tau(z) = (z-\tau)^{-1}\) (arising in frequency domain model order reduction), the exponential family \(f^\tau(z) = \exp(\tau z)\) (time domain), and the so-called \(\varphi\)-functions (exponential integrators). I will discuss some of the applications and report on recent work on the theory and computation of shared-denominator rational approximants.